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Chapter 34: Q25PE (page 1238)

Consider a supermassive black hole near the center of a galaxy. Calculate the radius of such an object based on its mass. You must consider how much mass is reasonable for these large objects, and which is now nearly directly observed. (Information on black holes posted on the Web by NASA and other agencies is reliable, for example.)

Short Answer

Expert verified

The radius of object based on it鈥檚 mass is obtained as: \({\rm{3 \times 1}}{{\rm{0}}^{{\rm{15}}}}{\rm{ m}}\).

Step by step solution

01

Schwarzschild radius

The Schwarzschild radius is given by,

\({R_s} = \frac{{2GM}}{{{c^2}}}\)

Here\({R_s}\)is the Schwarzschild radius,\(G\)is the universal gravitational constant,\(M\)is the mass of the black hole,\(c\)is the speed of the light.

02

Evaluating the radius

The mass of the supermassive black hole is:

\(M = {\rm{1}}{{\rm{0}}^{12}}{M_{{\rm{sun}}}}\)

Using the formula ofSchwarzschild to obtain the radius as:

\(\begin{align}{R_s}{\rm{ }} &= {\rm{ }}\frac{{2GM}}{{{c^2}}}\\ &= {\rm{ }}\frac{{{\rm{2 \times 6}}{\rm{.67 \times 1}}{{\rm{0}}^{{\rm{ - 11}}}}\frac{{{\rm{N}}{{\rm{m}}^{\rm{2}}}}}{{{\rm{k}}{{\rm{g}}^{\rm{2}}}}}{\rm{ \times 1}}{{\rm{0}}^{12}}{\rm{ \times 2 \times 1}}{{\rm{0}}^{{\rm{30}}}}{\rm{kg}}}}{{{{{\rm{(3 \times 1}}{{\rm{0}}^{\rm{8}}}\frac{{\rm{m}}}{{\rm{s}}}{\rm{)}}}^{\rm{2}}}}}{\rm{ }}\\ &= {\rm{ 3 \times 1}}{{\rm{0}}^{{\rm{15}}}}\,{\rm{m}}\end{align}\)

Therefore, the radius is \({\rm{3 \times 1}}{{\rm{0}}^{{\rm{15}}}}{\rm{ m}}\).

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